A352522 Triangle read by rows where T(n,k) is the number of integer compositions of n with k weak nonexcedances (parts on or below the diagonal).

1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 2, 3, 4, 3, 3, 1, 3, 4, 8, 6, 6, 4, 1, 4, 7, 12, 13, 12, 10, 5, 1, 5, 13, 16, 26, 24, 22, 15, 6, 1, 7, 19, 27, 43, 48, 46, 37, 21, 7, 1, 10, 26, 47, 68, 90, 93, 83, 58, 28, 8, 1, 14, 36, 77, 109, 159, 180, 176, 141

Offset: 0

Author

Gus Wiseman, Mar 22 2022

Keywords

• nonn
• tabl

Examples

			Triangle begins:
   1
   0   1
   1   0   1
   1   1   1   1
   1   3   1   2   1
   2   3   4   3   3   1
   3   4   8   6   6   4   1
   4   7  12  13  12  10   5   1
   5  13  16  26  24  22  15   6   1
   7  19  27  43  48  46  37  21   7   1
  10  26  47  68  90  93  83  58  28   8   1
For example, row n = 6 counts the following compositions:
  (6)   (15)   (114)  (123)   (1113)   (11112)  (111111)
  (24)  (42)   (132)  (1311)  (1122)   (11121)
  (33)  (51)   (141)  (2112)  (1131)   (11211)
        (231)  (213)  (2121)  (1212)   (12111)
               (222)  (2211)  (1221)
               (312)  (3111)  (21111)
               (321)
               (411)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
• MathOverflow, Why 'excedances' of permutations? [closed].

Crossrefs

Row sums are A011782.

The strong version for partitions is A114088.

The opposite version for partitions is A115994.

The version for permutations is A123125, strong A173018.

Column k = 0 is A238874.

The corresponding rank statistic is A352515.

The strong version is A352521, first column A219282, rank statistic A352514.

The strong opposite is A352524, first col A008930, rank statistic A352516.

The opposite version is A352525, first col A177510, rank statistic A352517.

A000041 counts integer partitions, strict A000009.

A008292 is the triangle of Eulerian numbers (version without zeros).

A238349 counts comps by fixed points, first col A238351, rank stat A352512.

A352488 lists the weak nonexcedance set of A122111.

A352523 counts comps by unfixed points, first A352520, rank stat A352513.

Cf. A088218, A098825, A238352, A352489.

Programs

Mathematica

pw[y_]:=Length[Select[Range[Length[y]],#>=y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pw[#]==k&]],{n,0,15},{k,0,n}]

PARI

T(n)={my(v=vector(n+1, i, i==1), r=v); for(k=1, n, v=vector(#v, j, sum(i=1, j-1, if(k>=i,x,1)*v[j-i])); r+=v); [Vecrev(p) | p<-r]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 19 2023